Chemically reactive and aging macromolecular mixtures I: Phase diagrams, spinodals, and gelation.

Multicomponent macromolecular mixtures often form higher-order structures, which may display non-ideal mixing and aging behaviors. In this work, we first propose a minimal model of a quaternary system that takes into account the formation of a complex via a chemical reaction involving two macromolecular species; the complex may then phase separate from the buffer and undergo a further transition into a gel-like state. We subsequently investigate how physical parameters such as molecular size, stoichiometric coefficients, equilibrium constants, and interaction parameters affect the phase behavior of the mixture and its propensity to undergo aging via gelation. In addition, we analyze the thermodynamic stability of the system and identify the spinodal regions and their overlap with gelation boundaries. The approach developed in this work can be readily generalized to study systems with an arbitrary number of components. More broadly, it provides a physically based starting point for the investigation of the kinetics of the coupled complex formation, phase separation, and gelation processes in spatially extended systems.

Growth kinetics of amyloid-like fibrils: An integrated atomistic simulation and continuum theory approach.

Amyloid fibrils have long been associated with many neurodegenerative diseases. The conventional picture of the formation and proliferation of fibrils from unfolded proteins comprises primary and secondary nucleation of oligomers followed by elongation and fragmentation thereof. In this work, we first employ extensive all-atom molecular dynamics (MD) simulations of short peptides to investigate the governing processes of fibril growth at the molecular scale. We observe that the peptides in the bulk solution can bind onto and subsequently diffuse along the fibril surface, which leads to fibril elongation via either bulk- or surface-mediated docking mechanisms. Then, to guide the quantitative interpretation of these observations and to provide a more comprehensive picture of the growth kinetics of single fibrils, a continuum model which incorporates the key processes observed in the MD simulations is formulated. The model is employed to investigate how relevant physical parameters affect the kinetics of fibril growth and identify distinct growth regimes. In particular, it is shown that fibrils which strongly bind peptides may undergo a transient exponential growth phase in which the entire fibril surface effectively acts as a sink for peptides. We also demonstrate how the relevant model parameters can be estimated from the MD trajectories. Our results provide compelling evidence that the overall fibril growth rates are determined by both bulk and surface peptide fluxes, thereby contributing to a more fundamental understanding of the growth kinetics of amyloid-like fibrils.

Nucleation and Growth of Amyloid Fibrils.

The formation of amyloid fibrils is a complex phenomenon that remains poorly understood at the atomic scale. Herein, we perform extended unbiased all-atom simulations in explicit solvent of a short amphipathic peptide to shed light on the three mechanisms accounting for fibril formation, namely, nucleation via primary and secondary mechanisms, and fibril growth. We find that primary nucleation takes place via the formation of an intermediate state made of two laminated ß-sheets oriented perpendicular to each other. The amyloid fibril spine subsequently emerges from the rotation of these ß-sheets to account for peptides that are parallel to each other and perpendicular to the main axis of the fibril. Growth of this spine, in turn, takes place via a dock-and-lock mechanism. We find that peptides dock onto the fibril tip either from bulk solution or after diffusing on the fibril surface. The latter docking pathway contributes significantly to populate the fibril tip with peptides. We also find that side chain interactions drive the motion of peptides in the lock phase during growth, enabling them to adopt the structure imposed by the fibril tip with atomic fidelity. Conversely, the docked peptide becomes trapped in a local free energy minimum when docked-conformations are sampled randomly. Our simulations also highlight the role played by nonpolar fibril surface patches in catalyzing and orienting the formation of small cross-ß structures. More broadly, our simulations provide important new insights into the pathways and interactions accounting for primary and secondary nucleation as well as the growth of amyloid fibrils.

Classification and Simulation of Structural Phase Transformation-Induced Interfacial Defects in Group VI Transition-Metal Dichalcogenide Monolayers.

Polymorphic 2D materials have recently emerged as promising candidates for use in nanoelectronic devices by way of their ability to undergo structural phase transformations induced by external fields. Under cyclic transformations, however, induced interfacial defects may proliferate and compromise the system properties. Herein, we first employ geometric analysis to classify such defects generated during the 2H ↔ 1T and 2H ↔ 1T' transformations in group VI transition-metal dichalcogenide monolayers. Then, simulations of a mesoscale model with atomistic spatial resolution are conducted to assess the proliferation of such defects during cyclic 2H ↔ 1T transformations. It is shown that defect densities reach a steady state, with the 2H phase remaining more pristine than the 1T and 1T' states. We expect that the effects of these defects on the device performance are application-dependent and will require further inquiry.

Anisotropic material depletion in epitaxial polymer crystallization.

The physical properties of a semicrystalline polymer thin film are intimately related to the morphology of its crystalline domains. While the mechanisms underlying crystallization of flat-on oriented polymer crystals are well known, similar mechanisms remain elusive for edge-on oriented thin films due to the propensity of substantially thin films to adopt flat-on orientations. Here, we employ an epitaxial polymer-substrate relationship to enforce edge-on crystallization in thin films. Using matrix-assisted pulsed laser evaporation (MAPLE), we deposit films in which crystal nucleation is spatially separated from subsequent epitaxial crystallization. These experiments, together with phase-field simulations, demonstrate a highly anisotropic and localized material depletion during edge-on crystallization. These results provide deeper insight into the physics of polymer crystallization under confinement and introduce a processing motif in the crystallization of ultrathin structured films.

Liquid-liquid phase separation within fibrillar networks.

Complex fibrillar networks mediate liquid-liquid phase separation of biomolecular condensates within the cell. Mechanical interactions between these condensates and the surrounding networks are increasingly implicated in the physiology of the condensates and yet, the physical principles underlying phase separation within intracellular media remain poorly understood. Here, we elucidate the dynamics and mechanics of liquid-liquid phase separation within fibrillar networks by condensing oil droplets within biopolymer gels. We find that condensates constrained within the network pore space grow in abrupt temporal bursts. The subsequent restructuring of condensates and concomitant network deformation is contingent on the fracture of network fibrils, which is determined by a competition between condensate capillarity and network strength. As a synthetic analog to intracellular phase separation, these results further our understanding of the mechanical interactions between biomolecular condensates and fibrillar networks in the cell.

Nucleation landscape of biomolecular condensates.

All structures within living cells must form at the right time and place. This includes condensates such as the nucleolus, Cajal bodies and stress granules, which form via liquid-liquid phase separation of biomolecules, particularly proteins enriched in intrinsically disordered regions (IDRs)1,2. In non-living systems, the initial stages of nucleated phase separation arise when thermal fluctuations overcome an energy barrier due to surface tension. This phenomenon can be modelled by classical nucleation theory (CNT), which describes how the rate of droplet nucleation depends on the degree of supersaturation, whereas the location at which droplets appear is controlled by interfacial heterogeneities3,4. However, it remains unknown whether this framework applies in living cells, owing to the multicomponent and highly complex nature of the intracellular environment, including the presence of diverse IDRs, whose specificity of biomolecular interactions is unclear5-8. Here we show that despite this complexity, nucleation in living cells occurs through a physical process similar to that in inanimate materials, but the efficacy of nucleation sites can be tuned by their biomolecular features. By quantitatively characterizing the nucleation kinetics of endogenous and biomimetic condensates in living cells, we find that key features of condensate nucleation can be quantitatively understood through a CNT-like theoretical framework. Nucleation rates can be substantially enhanced by compatible biomolecular (IDR) seeds, and the kinetics of cellular processes can impact condensate nucleation rates and specificity of location. This quantitative framework sheds light on the intracellular nucleation landscape, and paves the way for engineering synthetic condensates precisely positioned in space and time.

Evolution of Polymer Colloid Structure During Precipitation and Phase Separation.

Polymer colloids arise in a variety of contexts ranging from synthetic to natural systems. The structure of polymeric colloids is crucial to their function and application. Hence, understanding the mechanism of structure formation in polymer colloids is important to enabling advances in their production and subsequent use as enabling materials in new technologies. Here, we demonstrate how the specific pathway from precipitation to vitrification dictates the resulting morphology of colloids fabricated from polymer blends. Through continuum simulations, free energy calculations, and experiments, we reveal how colloid structure changes with the trajectory taken through the phase diagram. We demonstrate that during solvent exchange, polymer-solvent phase separation of a homogeneous condensate can precede polymer-polymer phase separation for blends of polymers that possess some degree of miscibility. For less-miscible, higher-molecular-weight blends, phase separation and kinetic arrest compete to determine the final morphology. Such an understanding of the pathways from precipitation to vitrification is critical to designing functional structured polymer colloids.

Defect-Enabled Phase Programming of Transition Metal Dichalcogenide Monolayers.

The ability to tune the local electronic transport properties of group VI transition metal dichalcogenide (TMD) monolayers by strain-induced structural phase transformations ("phase programming") has stimulated much interest in the potential applications of such layers as ultrathin programmable and dynamically switchable nanoelectronics components. In this manuscript, we propose a new approach toward controlling TMD monolayer phases by employing macroscopic in-plane strains to amplify heterogeneous strains arising from tailored, spatially extended defects within the monolayer. The efficacy of our proposed approach is demonstrated via numerical simulations of emerging domains localized around arrays of holes, grain boundaries, and compositional heterointerfaces. Quantitative relations between the macroscopic strains required, spatial resolution of domain patterns, and defect configurations are developed. In particular, the introduction of arrays of holes is identified as the most feasible phase programming route.

Strain Relaxation in Misfitting Transition Metal Dichalcogenide Monolayer Superlattices: Wrinkling vs Misfit Dislocation Formation.

Two-dimensional (2D) superlattices composed of chemically heterogeneous transition-metal dichalcogenides (TMDs) have been proposed as key components in next-generation optoelectronic devices. For potential applications, coherent, defect-free compositional interfaces are usually required. In this paper, a combination of scaling theory and numerical analysis is employed to investigate strain relaxation mechanisms in misfitting, chemically heterogeneous TMDs. We demonstrate that, in free-standing superlattices, wrinkling of the monolayer is asymptotically preferred over misfit dislocation formation in both binary and ternary superlattices. For substrate-supported monolayers, however, misfit dislocation formation is thermodynamically favored above a critical superlattice width, implying the presence of an upper limit to the thermodynamic stability of coherent, misfitting 2D superlattices. Finally, it is shown numerically that the critical superlattice width is only weakly dependent on the misfit.

Phase behavior and morphology of multicomponent liquid mixtures.

Multicomponent systems are ubiquitous in nature and industry. While the physics of few-component liquid mixtures (i.e., binary and ternary ones) is well-understood and routinely taught in undergraduate courses, the thermodynamic and kinetic properties of N-component mixtures with N > 3 have remained relatively unexplored. An example of such a mixture is provided by the intracellular fluid, in which protein-rich droplets phase separate into distinct membraneless organelles. In this work, we investigate equilibrium phase behavior and morphology of N-component liquid mixtures within the Flory-Huggins theory of regular solutions. In order to determine the number of coexisting phases and their compositions, we developed a new algorithm for constructing complete phase diagrams, based on numerical convexification of the discretized free energy landscape. Together with a Cahn-Hilliard approach for kinetics, we employ this method to study mixtures with N = 4 and 5 components. We report on both the coarsening behavior of such systems, as well as the resulting morphologies in three spatial dimensions. We discuss how the number of coexisting phases and their compositions can be extracted with Principal Component Analysis (PCA) and K-means clustering algorithms. Finally, we discuss how one can reverse engineer the interaction parameters and volume fractions of components in order to achieve a range of desired packing structures, such as nested "Russian dolls" and encapsulated Janus droplets.

Liquid Nuclear Condensates Mechanically Sense and Restructure the Genome.

Phase transitions involving biomolecular liquids are a fundamental mechanism underlying intracellular organization. In the cell nucleus, liquid-liquid phase separation of intrinsically disordered proteins (IDPs) is implicated in assembly of the nucleolus, as well as transcriptional clusters, and other nuclear bodies. However, it remains unclear whether and how physical forces associated with nucleation, growth, and wetting of liquid condensates can directly restructure chromatin. Here, we use CasDrop, a novel CRISPR-Cas9-based optogenetic technology, to show that various IDPs phase separate into liquid condensates that mechanically exclude chromatin as they grow and preferentially form in low-density, largely euchromatic regions. A minimal physical model explains how this stiffness sensitivity arises from lower mechanical energy associated with deforming softer genomic regions. Targeted genomic loci can nonetheless be mechanically pulled together through surface tension-driven coalescence. Nuclear condensates may thus function as mechano-active chromatin filters, physically pulling in targeted genomic loci while pushing out non-targeted regions of the neighboring genome. VIDEO ABSTRACT.

Physical principles of intracellular organization via active and passive phase transitions.

Exciting recent developments suggest that phase transitions represent an important and ubiquitous mechanism underlying intracellular organization. We describe key experimental findings in this area of study, as well as the application of classical theoretical approaches for quantitatively understanding these data. We also discuss the way in which equilibrium thermodynamic driving forces may interface with the fundamentally out-of-equilibrium nature of living cells. In particular, time and/or space-dependent concentration profiles may modulate the phase behavior of biomolecules in living cells. We suggest future directions for both theoretical and experimental work that will shed light on the way in which biological activity modulates the assembly, properties, and function of viscoelastic states of living matter.

Lipid Domain Co-localization Induced by Membrane Undulations.

Multicomponent lipid bilayer membranes display rich phase transition and associated compositional lipid domain formation behavior. When both leaflets of the bilayer contain domains, they are often found co-localized across the leaflets, implying the presence of a thermodynamic interleaflet coupling. In this work, it is demonstrated that fluctuation-induced interactions between domains embedded within opposing membrane leaflets provide a robust means to co-localize the domains. In particular, it is shown via a combination of a mode-counting argument, a perturbative calculation, and a non-perturbative treatment of a special case, that spatial variations in membrane bending rigidity associated with lipid domains embedded within the background phase always lead to an attractive interleaflet coupling with a magnitude of ∼0.01kBT/nm2 in simple model membrane systems. Finally, it is demonstrated that the fluctuation-induced coupling is very robust against membrane tension and substrate interactions.

Dynamic Phase Engineering of Bendable Transition Metal Dichalcogenide Monolayers.

Current interest in two-dimensional (2D) materials is driven in part by the ability to dramatically alter their optoelectronic properties through strain and phase engineering. A combination of these approaches can be applied in quasi-2D transition metal dichalcogenide (TMD) monolayers to induce displacive structural transformations between semiconducting (H) and metallic/semimetallic (T') phases. We classify such transformations in Group VI TMDs, and formulate a multiscale, first-principles-informed modeling framework to describe evolution of microstructural domain morphologies in elastically bendable 2D monolayers. We demonstrate that morphology and mechanical response can be controlled via application of strain either uniformly or through local probes to generate functionally patterned conductive T' domains. Such systems form dynamically programmable electromechanical 2D materials, capable of rapid local switching between domains with qualitatively different transport properties. This enables dynamic "drawing" of localized conducting regions in an otherwise semiconducting TMD monolayer, opening several interesting device-relevant functionalities such as the ability to dynamically "rewire" a device in real time.

Spatiotemporal Control of Intracellular Phase Transitions Using Light-Activated optoDroplets.

Phase transitions driven by intrinsically disordered protein regions (IDRs) have emerged as a ubiquitous mechanism for assembling liquid-like RNA/protein (RNP) bodies and other membrane-less organelles. However, a lack of tools to control intracellular phase transitions limits our ability to understand their role in cell physiology and disease. Here, we introduce an optogenetic platform that uses light to activate IDR-mediated phase transitions in living cells. We use this "optoDroplet" system to study condensed phases driven by the IDRs of various RNP body proteins, including FUS, DDX4, and HNRNPA1. Above a concentration threshold, these constructs undergo light-activated phase separation, forming spatiotemporally definable liquid optoDroplets. FUS optoDroplet assembly is fully reversible even after multiple activation cycles. However, cells driven deep within the phase boundary form solid-like gels that undergo aging into irreversible aggregates. This system can thus elucidate not only physiological phase transitions but also their link to pathological aggregates.

Transbilayer Colocalization of Lipid Domains Explained via Measurement of Strong Coupling Parameters.

When micron-scale compositional heterogeneity develops in membranes, the distribution of lipids on one face of the membrane strongly affects the distribution on the other. Specifically, when lipid membranes phase separate into coexisting liquid phases, domains in each monolayer leaflet of the membrane are colocalized with domains in the opposite leaflet. Colocalized domains have never been observed to spontaneously move out of registry. This result indicates that the lipid compositions in one leaflet are strongly coupled to compositions in the opposing leaflet. Predictions of the interleaflet coupling parameter, Λ, vary by a factor of 50. We measure the value of Λ by applying high shear forces to supported lipid bilayers. This causes the upper leaflet to slide over the lower leaflet, moving domains out of registry. We find that the threshold shear stress required to deregister domains in the upper and lower leaflets increases with the inverse length of domains. We derive a simple, closed-form expression relating the threshold shear to Λ, and find Λ = 0.016 ± 0.004 kBT/nm2.

Simulation study of twisted crystal growth in organic thin films.

Many polymer and organic small-molecule thin films crystallize with microstructures that twist or curve in a regular manner as crystal growth proceeds. Here we present a phase-field model that energetically favors twisting of the three-dimensional crystalline orientation about and along particular axes, allowing morphologies such as banded spherulites, curved dendrites, and "s"- or "c"-shaped needle crystals to be simulated. When twisting about the fast-growing crystalline axis is energetically favored and spherulitic growth conditions are imposed, crystallization occurs in the form of banded spherulites composed of radially oriented twisted crystalline fibers. Due to the lack of symmetry, twisting along the normal growth direction leads to heterochiral banded spherulites with opposite twist handedness in each half of the spherulite. When twisting is instead favored about the axis perpendicular to the plane of the substrate and along the normal growth direction under diffusion-limited single-crystalline growth conditions, crystallization occurs in the form of curved dendrites with uniformly rotating branches. We show that the rate at which the branches curve affects not only the morphology but also the overall kinetics of crystallization, as the total crystallized area at a given time is maximized for a finite turning rate.

RNA transcription modulates phase transition-driven nuclear body assembly.

Nuclear bodies are RNA and protein-rich, membraneless organelles that play important roles in gene regulation. The largest and most well-known nuclear body is the nucleolus, an organelle whose primary function in ribosome biogenesis makes it key for cell growth and size homeostasis. The nucleolus and other nuclear bodies behave like liquid-phase droplets and appear to condense from the nucleoplasm by concentration-dependent phase separation. However, nucleoli actively consume chemical energy, and it is unclear how such nonequilibrium activity might impact classical liquid-liquid phase separation. Here, we combine in vivo and in vitro experiments with theory and simulation to characterize the assembly and disassembly dynamics of nucleoli in early Caenorhabditis elegans embryos. In addition to classical nucleoli that assemble at the transcriptionally active nucleolar organizing regions, we observe dozens of "extranucleolar droplets" (ENDs) that condense in the nucleoplasm in a transcription-independent manner. We show that growth of nucleoli and ENDs is consistent with a first-order phase transition in which late-stage coarsening dynamics are mediated by Brownian coalescence and, to a lesser degree, Ostwald ripening. By manipulating C. elegans cell size, we change nucleolar component concentration and confirm several key model predictions. Our results show that rRNA transcription and other nonequilibrium biological activity can modulate the effective thermodynamic parameters governing nucleolar and END assembly, but do not appear to fundamentally alter the passive phase separation mechanism.